I. INTRODUCTION devices [25].

The Analytic Hierarchy Process (AHP) method

stands out as the favored approach in Multiple Criteria Decision
The electrical power system faces several challenges such as Making (MCDM) due to its notable advantages in terms of
voltage sags and excessive power flow in some branches, which computational efficiency and user-friendly comprehensibility
result from the mismatch between power generation and load [26].
demand, the lack of reactive power supply, or sudden
disturbances in transmission lines. These factors may ultimately Most studies focus on improving voltage stability and reducing
lead to a total blackout [1,2]. To overcome these challenges, the generation cost, voltage deviation, installation cost of the FACTS
Power Electronics Technology has enabled the development of devices, and real power loss in transmission line [25, 27-29].
Flexible AC Transmission Systems (FACTS), which are power- Various voltage stability indices aid in identifying optimal SVC
electronic controllers that enhance the controllability and stability locations. However, the Line Stability Index’s (Lmn) limitation
of the network. FACTS controllers are widely used for voltage lies in its inability to predict collapse under very high active or
control, as well as for reducing losses, regulating load flow, apparent power loading, attributed to its disregard for active
enhancing transient stability and mitigating harmonics [3]. power effects and shunt admittance, which are also disregarded
Among the FACTS devices, the shunt ones, such as shunt in the Fast Voltage Stability Index (FVSI) along with angle
capacitor, SVC and STATCOM, can regulate the reactive power differences [30]. Consequently, despite its rapid calculation, the
flow in the network by injecting or absorbing capacitive or FVSI lacks accuracy due to its exclusion of active power
inductive reactive power, thereby stabilizing voltage [4]. Shunt variations and angle differences. The formulation of the Line
VAR compensators (SVC) are predominantly used for this Stability Factor (Lqp) index entirely disregards line resistance,
purpose as they are simple, cost effective and easy to install and resulting in inaccurate collapse predictions and neglecting the
operate. However, SVC and STATCOM have better performance relative direction of active power flow with respect to reactive
than a simple shunt capacitor in terms of reducing losses and power flow. To overcome this limitation, a novel collapse
improving the voltage profile, but they also have higher costs. prediction index (NCPI) was proposed [31], but it had a higher
computational time than FVSI. In a previous study, a method was
Optimal location of SVC devices is an important problem in used to assess the economic viability of installing SVC devices in
power system planning and operation, as it can improve various the power network by calculating the return on investment (ROI)
performance indicators, such as voltage profile, power loss, and payback period. This involves considering factors such as
economic benefit, and others. Therefore, finding the optimal loan amount, interest rate, and savings in generation costs [16].
location of the FACTS device is crucial. Several heuristic and
meta-heuristic optimization techniques have been applied to this This paper presents a hybrid algorithm of Artificial Bee Colony
problem, such as genetic algorithm (GA) [5,6], particle swarm (ABC) and Particle Swarm Optimization (PSO) for optimal
optimization (PSO) [6,7], artificial bee colony (ABC) [8], multi- placement of SVC in the standard IEEE 14-bus, and 30-bus test
objective cuckoo search algorithm (MOCS) [9], and cuckoo systems under increased load conditions. The optimal location of
search algorithm (CS) [10]. This integration results in increased SVC is determined by using a modified voltage stability index
stability of parameters, faster convergence, improved solution called the modified collapse prediction index (MCPI), which
quality, and reduced computation time [11,12]. Some examples integrates the advantages of two existing indices: the FVSI and
of hybrid algorithms for optimal location of SVC devices are: the NCPI. The optimization problem is formulated with four
particle swarm optimization (PSO) with gravitational search objective functions, with the criteria weights derived via the
algorithm (GSA) [13], artificial bee colony (ABC) algorithm with Analytic Hierarchy Process (AHP). Furthermore, a methodology
sequential quadratic programming (SQP) [14], and artificial bee for calculating the recovery time based on the investment made
colony (ABC) with differential evolution (DE) [15], cuckoo in acquiring the SVC is incorporated into the scheme.
search algorithm (CSA) with chemical reaction optimization
(CRO) [16]. II. PROBLEM FORMULATION
2.1 MODELLING OF SVC
One of the challenges in multi-objective optimization of FACTS
devices is to select appropriate weights that reflect the relative The Static Var Compensator (SVC) serves as a shunt-connected
importance of different objectives. The literature shows various static reactive power source, comprising a capacitor and
methods and approaches for determining the weight coefficients a thyristor-controlled reactor [32]. When connected in shunt with
of the criteria. Some studies used equal weights or approximate the transmission line through a shunt transformer (as depicted in
values obtained by error method [17-19]. Some studies relied on Fig. 1) [33], the SVC behaves akin to a variable reactance,
the decision maker’s preferences for assigning the weights [20, capable of providing either inductive or capacitive
21]. Some studies experimented with different combinations of compensation. Its primary function is to inject or absorb reactive
equal and preference-based weights according to different power at the connected bus, thereby controlling the voltage of that
scenarios [22-24]. In a previous study, an Analytic Hierarchy bus [34]. The reactive power and current injected into bus k can
Process (AHP) approach, which is one of the Multi-Criteria be calculated using the following expressions:
Decision Making (MCDM) methods, was applied to determine 𝐼𝑆𝑉𝐶 = 𝑗𝑉𝑘 × 𝐵𝑆𝑉𝐶 (1)
these weights in finding the optimal location of various FACTS 𝑄𝑆𝑉𝐶 = 𝑉𝑘2 × 𝐵𝑆𝑉𝐶 (2)
 from thorough experimentation on a sizable dataset. Table 2
ISVC and QSVC represent the injected or absorbed reactive current presents the randomness index values for various n values. A
and reactive power of SVC. BSVC is the SVC's susceptance, and pair-wise comparison is deemed consistent if the CR values are
Vk is the voltage of the bus to which the SVC is connected [35]. below 10%. Conversely, if the CR value exceeds 10%, indicating
inconsistency, weights are re-evaluated and adjusted in the pair-
wise comparison matrix.
TABLE 2. Average random consistency (RI). [36].

Size of 3 4 5 6 7 8 9 10
matrix
Random 0.58 0.9 1.12 1.24 1.32 1.41 1.45 1.49
consistency

2.3 OBJECTIVE FUNCTIOS

FIGURE 1. Functional diagram of SVC
A multi-objective optimal model has been formulated to ascertain
2.2 ANALYTIC HIERARCHY PROCESS (AHP) the optimal location and capacity of the SVC within power
The Analytic Hierarchy Process serves as a structured decision- systems. The model is designed to minimize various factors
making tool essential for dissecting complex problems into including voltage stability index, voltage deviation, installation
manageable criteria. It operates on three fundamental principles: costs associated with SVC deployment, and real power losses
problem decomposition, comparative judgment, and synthesis of across transmission lines. These considerations are integrated
relative importance or rankings [36, 37]. Within the AHP into the proposed objective function, enhancing the efficiency
framework, complex problems are hierarchically structured into and reliability of power system operations.
criteria, facilitating a systematic comparison among them. This 1) MINIMIZATION OF VOLTAGE STABILITY INDEX
comparative evaluation, termed pair-wise comparison, is pivotal
in deriving rankings. Subsequently, the Eigen vector method is The identification of the ideal location for the SVC is achieved
employed to compute these rankings, followed by a consistency through our innovative approach known as the Modified Collapse
check using the consistency ratio [36]. Notably, Table 1 Prediction Index (MCPI). This method is designed to pinpoint the
delineates the scale of pairwise comparison as defined by Saaty. weakest bus within the power system, a crucial step in optimizing
TABLE 1. Scale of Analytic Hierarchy Process (AHP) [36]. the placement of the SVC to enhance voltage stability. MCPI
index can be explained from Fig. 2.
Degree of Preference Definition
1 Equally Important
3 Moderately Important
5 Highly Important
7 Very Highly Important .
9 Extremely Important FIGURE 2. Two bus system
2,4,6,8 Intermediate Values
4𝑍 4 𝑃𝑟2
To ensure the reliability of the relative importance weights 𝑀𝐶𝑃𝐼 = 𝐹𝑉𝑆𝐼 × 𝛽 + (𝐹𝑉𝑆𝐼 + ) × (𝛽 − 1)
𝑋 2 𝑉𝑠4 (5)
determined through pair-wise comparisons, it is essential to
assess their consistency using the following equation: 𝑂𝐹1 = 𝑀𝐶𝑃𝐼 = 𝐹𝑉𝑆𝐼 × 𝛽 + 𝑀𝐶𝑃𝐼 × (𝛽 − 1) (6)

Consistency Ratio, 𝐶𝑅 =
𝐶𝐼
(3) Where, Vr and Vs denote the sending and receiving voltages,
𝑅𝐼
respectively, with δ representing the voltage angle disparity
In this equation, CI represents the consistency index, while RI between the sending δs and receiving δr angles. Ps and Qs signify
denotes the randomness index. The calculation of CI is as the active and reactive power at the sending terminal, while Pr
follows: and Qr symbolize their counterparts at the receiving terminal. The
𝜆𝑚𝑎𝑥 −𝑛 (4) switching function 𝛽 of the voltage angle difference δ is defined
Consistency index, 𝐶𝐼 =
𝑛−1 as follows:
Here, 𝜆𝑚𝑎𝑥 refers to the major eigenvalue, and n signifies the 1 𝛿 < 𝛿𝐶
order of the matrix. 𝛽={ (7)
0 𝛿 ≥ 𝛿𝐶
The values of the randomness index provided by Saaty are To determine β, we examine the error percentage of voltage
dependent on the order of the matrix, denoted by n. RI is derived stability indices concerning the voltage angle difference δ [38],
utilizing the base case results from NCPI and FVSI. Error EMI
τ = log (1+r) [ ] (14)
percentage is determined using the relative error of the EMI − Csvc × r
approximation, which can be mathematically represented as:
Actual Value − Measured Value
Where, EMI stands for the Equated Monthly Installment and
Percentage Error, % E = | | (8) r represents the monthly interest rate. The monthly economic
Actual Value
benefit obtained from generation cost savings is to be paid as EMI
During error calculation, the NCPI is recognized as the actual till the recovery time τ is reached. The annual interest rate is
value and is esteemed as the most precise representation of the considered to be 10% so the monthly rate r is calculated to be
voltage stability index. Conversely, the FVSI is utilized as the 0.8333%.
measured value in this comparison. The MCPI index assumes
values between 0 and 1, where a value closer to 1 designates the 2.5 FITNESS FUNCTION
weakest transmission line, with the associated bus being
identified as the weakest node within the system. Our objective in this case is to reduce all criteria, which we will
do by applying our suggested approach to minimize the fitness
2) MINIMIZATION OF ACTIVE POWER LOSS function. We normalize the data into a range between 0 and 1
to ensure comparability across several criteria with differing
In addition to ensuring voltage stability, minimizing active power value ranges. When the SVC is positioned at the n th vulnerable
losses (𝑃𝑙𝑜𝑠𝑠 ) is essential from a financial perspective [39]. Active bus, the fitness function is represented as 𝐹𝑛 .
power losses (𝑃𝑙𝑜𝑠𝑠 ) can be represented as follows:
𝑀𝐶𝑃𝐼sum 𝑃𝑙𝑜𝑠𝑠
𝑁 𝐹𝑛 = ω1 + ω2
max{MCPI𝑠𝑢𝑚 } 𝑚𝑎𝑥{𝑃𝑙𝑜𝑠𝑠 } (15)
𝑂𝐹2 = 𝑃𝑙𝑜𝑠𝑠 = ∑ 𝑔𝑚 [𝑉𝑠2 + 𝑉𝑟2 − 2𝑉𝑠 𝑉𝑟 cos(𝛿𝑠 − 𝛿𝑟 )] (9) 𝑉𝐷 𝐶𝑇
𝑚=1 + 𝜔3 + 𝜔4
𝑚𝑎𝑥{𝑉𝐷} 𝑚𝑎𝑥{𝐶𝑇 }
where, 𝑔𝑚 is the conductance of transmission line and N is the
where ω1 , ω2 , ω3 and ω4 represent the respective coefficients of
number of buses.
preference assigned to different criteria. It is essential to note that
3) MINIMIZATION OF THE VOLTAGE DEVIATION the sum of these coefficients must equal to one.
4

Improving the quality of power in a power system requires ∑ ω𝑚 = 1 (16)

keeping voltage profiles within specific limits. We achieve this m=1
by reducing voltage deviation, which is done using the equation In the process of determining weighting factors for multi-
below: objective optimization, careful consideration is given to the
𝑁
(10) preferences of the decision maker. In this particular case study,
𝑂𝐹3 = 𝑉𝐷 = ∑ |1 − 𝑉𝑚 |
priority has been assigned to voltage stability as the primary
𝑚=1
focus. The MCPI index holds the utmost importance in
Here, 𝑉𝑚 represents the voltage magnitude at bus m. In this influencing voltage stability, with voltage deviation, active power
investigation, an acceptable bus voltage range of 0.95 to 1.05 per losses, and SVC installation cost having comparatively lesser
unit (p.u.) is adopted. impact. Detailed insights into the relative importance of these
criteria can be found in the pair-wise comparison matrix provided
4) MINIMIZATION OF COST FUNCTION
in Table 3. The coefficients (weights) ω1 , ω2 , ω3 and ω4 are
The total expenditure involved in installing SVC encompasses calculated by AHP method to 0.579, 0.2261, 0.1434 and 0.0515
both the initial investment in hardware equipment and ongoing respectively.
maintenance costs [16]. This overall cost is minimized through
the following equation: TABLE 3. Pair-wise comparison matrix of the criteria.
MCPI VD 𝑷𝒍𝒐𝒔𝒔 𝑪𝑻
𝑂𝐹4 = 𝐶𝑇 = 𝐶𝑆𝑉𝐶 + 𝐶𝑚𝑚𝑡 (11)
MCPI 1 3 5 7
Here, the cost of SVC installation (in dollars) is denoted by 𝐶𝑆𝑉𝐶 , VD 1/3 1 2 6
calculated as:
𝑷𝒍𝒐𝒔𝒔 1/5 1/2 1 4
𝐶𝑆𝑉𝐶 = (0.0003𝑆 2 − 0.3051𝑆 + 127.38) × 1000 (12) 𝑪𝑻 1/7 1/6 1/4 1
Furthermore, the maintenance cost (𝐶𝑚𝑚𝑡 ) of the SVC is
estimated to be 5% of the installation cost [29], as expressed by The fitness function 𝐹𝑛 reaches its peak when no SVC is
Equation (13): connected to the system, gradually decreasing as the system's
performance improves with the connection of the SVC at
𝐶𝑚𝑚𝑡 = 0.05 × 𝐶𝑆𝑉𝐶 (13) different locations. The minimum value of the fitness function
2.4 RECOVERY TIME FUNCTION indicates the optimal system response when the SVC is connected
to a specific vulnerable bus n.
To determine the recovery time (τ) for the investment in SVC,
interest on the principal loan amount must be taken into
consideration, as shown in Equation (14) [16].
III. PROPOSED APPROACH Step 6: Choose the new food source and its variable using an "if-
else" statement.
The best location for the SVC is found by combining PSO's
velocity and position calculation with the ABC algorithm. In the 3.3 ONLOOKER BEE PHASE
ABC algorithm, employed bees exploit food sources, while
unemployed bees (including scout and onlooker bees) search The onlooker bees employ a probabilistic method to assess the
randomly for food and wait for information from employed bees. insights gathered by the employed bees. A higher probability
Meanwhile, the PSO algorithm updates its initial particles by increases the likelihood of selecting a new food source, while
searching for the best solutions and adjusting its entire those with lower probabilities may be disregarded. This
population. This hybrid ABC-PSO algorithm follows these steps evaluation process determines the suitability of each food source
[40]: in contributing to the quest for the optimal solution.

3.1 INITIALIZATION OF DATA SETS Step 7: The onlooker bee conducts a probability assessment using
the following equation:
The initialization phase of the ABC algorithm serves to establish 𝑓𝑖𝑡(𝑥𝑖 )
the data variables essential for optimization. These variables are 𝑃𝑖 = 𝑠𝑛 (19)
𝛴𝑖 𝑓𝑖𝑡(𝑥𝑖 )
defined within their designated minimum and maximum limits to Here, 𝑓𝑖𝑡(𝑥𝑖 ) indicates the suitability of the ith food source
ensure the optimization process remains within feasible bounds. relative to the objective function, while Sn denotes the population
size of the food sources.
Step 1: Random initialization of food sources is conducted based
on the upper and lower bounds of the location (bus1 to bus14) for Step 8: If the food source chosen by the onlooker bee proves to
the IEEE 14-bus system and (bus1 to bus30) for the IEEE 30-bus be suitable and superior to the previous variable, the employed
system, along with the size of SVC (ranging from 5 to 50 bee adopts this new position and resets the trial counter.
MVAR). The number of food sources is set at 50, constituting Otherwise, the trial counter increments by 1.
half the colony size (CS = 100 bees), and the problem dimension
Step 9: Subsequently, the onlooker bee randomly computes the
is defined as 14 buses for the IEEE 14-bus system and 30 buses neighbor of the selected food source, evaluates its fitness, and
for the IEEE 30-bus system. The base apparent power is reapplies the step 6.
standardized at 100 MVA. Additionally, parameters for velocity
computation in the PSO algorithm are initialized as follows: 3.4 SCOUT BEE PHASE
Coefficients c1 and c2 are set to 2, while the inertia weight wmax is
assigned 1 and wmin is set to 0.1 [40]. This phase is enacted depending on the number of solutions that
can be improved, and it is initiated only when the probability
Step 2: The iteration commences with τ=0, with the process value of the food source fails to improve beyond a predetermined
limited to 100 iterations to facilitate effective progression through threshold. Additionally, only one scout bee phase can take place
the optimization procedure. during each iteration of the ABC algorithm.

3.2 EMPLOYED BEE PHASE Step 10: A scout bee is tasked with randomly searching for the
next food source, without considering past occurrences, and
During this phase, the employed bees carefully explore nearby incrementing the iteration counter (τ = τ + 1) in incremental
areas to find places close to their current food sources. If they find steps.
a better spot, they switch to it. Unlike the PSO approach, where
all positions undergo simultaneous updates, each employed bee 3.5 HYBRID OF THE ALGORITHM
iteratively updates only one food source, following these steps:
The 5th phase of the ABC algorithm marks the Termination Phase.
Step 3: Assess the fitness of the food source using Equation (15). However, in phase 2, the PSO algorithm replaces Equations (17)
and (18) with Equations (20) and (21):
Step 4: Identify parameters to undergo random adjustments,
ensuring that the selected neighboring food source differs from vnew (t) = w1 v𝑜𝑙𝑑 (t − 1) + 𝑐1 𝑟1 (𝑋best − 𝑋𝑜𝑙𝑑 )
the current one. (20)
+ 𝑐2 𝑟2 (Gbest − 𝑋𝑜𝑙𝑑 )
Step 5: In the ABC algorithm, utilize Equations (17) and (18) to 𝑋new (t) = 𝑋𝑜𝑙𝑑 (t − 1) + 𝑣𝑛𝑒𝑤 (𝑡) (21)
compute the velocity and update the position of the food source:
Here, vnew and 𝑋new represent the velocity and position of the
𝜈𝑖,𝑗 = 𝑥𝑖,𝑗 + 𝜑𝑖,𝑗 (𝑥𝑖,𝑗 − 𝑥𝑘,𝑗 ) (17) new food source respectively. The inertia weight w (1) maintains
a balance between global and local search abilities in PSO. The
𝑥𝑖,𝑑 = 𝑥𝑑𝑚𝑖𝑛 + 𝑟(𝑥𝑑𝑚𝑎𝑥 − 𝑥𝑑𝑚𝑖𝑛 ) (18) acceleration coefficients 𝑐1 and 𝑐2 (2) influence the maximum
step size in each iteration. The variables 𝑟1 and 𝑟2 are random
In these equations, i is the employed bee, j is a random index numbers ranging between 0 and 1, contributing to the stochastic
based on the problem size, k is a random neighboring index to i, nature of the algorithm. 𝑋best , represents the best position of the
φ is a random number between 0 and 1, r is a real number between food source, while Gbest is the global best position of the food
0 and 1, and d shows the problem's size with specific minimum source across the entire population. It's worth noting that the
and maximum values.
 global best solution isn't directly utilized in the ABC algorithm to
determine a new food position [40].

3.6 THE STOPPING CRITERIA

Step 11: This process relies on the maximum number of iterations
and the predetermined limit, which are determined from the
employed bees (Food Number) and problem dimension. It
involves updating the best food source and particle fitness [40].
Figure 3 illustrates the flow chart of the hybrid algorithm
employed in this study.
All data utilized in this study are acquired from power flow
analysis conducted in the MATLAB (R2023a) command prompt.
The tests were performed on an Intel Core i5 processor with the
following specifications: 2.4GHz CPU, 8GB RAM.

IV. RESULTS AND DISCUSSION

4.1 IEEE 14-BUS TEST SYSTEM
The optimal placement of SVC in the IEEE 14-bus system with
five connected generators at bus locations 1, 2, 3, 6 and 8, shown
in Fig. 4, begins with identifying the vulnerable bus locations.
The bus data, line data and generator coefficients are obtained
from [33,34] and the SVC has a range of 30 MVAr. The MCPI
values for the various transmission lines for normal load
conditions are shown in fig. 5. From fig. 5, the vulnerability is
observed at the lines 1–5, 4–9 and 5–6 at normal load.

Hence, it is concluded that the bus number 9, 5, 6 and 4 are the

vulnerable buses of the system. The SVC is placed at bus numbers
9, 5 and 4 individually and the obtained results are given in Table
2.
FIGURE 3. Flow chart of the hybrid ABC-PSO algorithm.

FIGURE 4. Standard IEEE 14-bus system

FIGURE 5. MCPI values for normal load conditions

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